I understand where you're coming from, but there;s little you can do. Some integrals can be done, some not. In your place, I'd concentrate on understanding how the Kramers recurrings can be derived without resorting to complicated integrals. Do you have a source which treats this thoroughly ?

Well the Zettili's book gives solved problem, but I still have problems understanding as to why she derived the radial equation with orbital quantum number, or electric charge :\

I saw it only now. It was just a trick. Admittedly, I haven't seen it in any other source, it's very interesting, indeed. So this trick with the 'l' differentiation is neat, you have to take it for granted. It gives you <1/r^2> and the one next with the <e> differentiation gives you the <1/r>.

dingo_d said:

I thought that once you have the recurrence relation the obtaining the formula should be straight forward :\

I tried with partial integration but it doesn't give good answer...

They do, it's just a recurring relation, so it suffices to know the average for 2 powers of r and you'll get it for all other, negative or positive.

For the <1/r> one could also use the virial theorem.

So the problems 6.2 & 6.3 of Zettili indeed solve a lot of issues, if you're able to prove Kramer's relation all by its own.

Zettili of course doesn't do it, which is the only drawback of this chapter 6.

I just checked Constantinescu & Magyari problem book (Pergamon Press, 1971) on QM and saw the Kramers equation and the trick with the differentiation wrt l in problems 6 and 7 of chapter 3. Indeed the proof of Kramers one is difficult using integrals.